What are daily obligations during the sprint of Product Owner?

It depends on the product and organization context, but in my situation (oversimplifying it), it mostly involve some kind of those:

• Product Backlog management
• Key stakeholders management (which also include C-levels')
• Analysis of current product utilization, usage and problems faced by users (customers)
• Analysis of current market situation and our competition
• Working closely with the dev team which mostly involve him into:
  • Making business decision that is needed to be made during current development in sprint
  • Product Backlog Refinement activity

The Product Owner has many stances and roles.  Not always doing everything, but ultimately responsible for  ensuring it all happens.   Check out this blog on the stances of the Product Owner.  

I talk to my clients and end-users almost everyday. Even if the Development Team has selected a refined item to work on, I will ask my client if that item is still valuable to them. I am also engaged with the Development Team to see if the work selected for the Sprint will be Done. And if not, I negotiate with them on other items they could work on which are refined and will meet the Sprint Goal. Besides the customer, I also need to talk to the Business to see if the Product will meet the Company's vision and goals. All of these are always ongoing. 

Just yesterday, a leader at the client business asked me when the work for the Product would be done. This leader is well-versed in traditional project management and was expecting me to say, "in two years, work on this project will be done." Instead, I told her that as long as the Product is delivering value to her, it will continue to emerge.

Stephanie Ockerman writes about emergence here:

Emergence implies that all solutions to all problems will become clear as we work, not simply by talking about them.  Essentially, we need the ability to learn for experience and respond to what we are learning.  This could mean fine-tuning our direction, changing course altogether, or continuing forward as our assumptions are validated.

This is an interesting question, as I see the same questions popping up a lot, but regarding the Scrum Master instead of a Product Owner. Outside of the comments above, I'd say talk to an actual Product Owner. Also Product Mastery by Geoff Watts and The Professional Product Owner by Ralph Jocham and Don McGreal are great books to read about this.

If I have to put it in two words, I'd say "Active Listening". Whatever stance or role the PO needs to play, the basis of it should be active listening. POs that spend enough time on developing this fundamental skill have access and ability to implement all five fundamental Scrum values within and outside Scrum team/s.  The Scrum Master facilitates and should be the point-man to help him hone on this skill. 

I have worked with a few Product Owners that spent a lot of time in conversations with stakeholders learning about their day to day operations, finding areas in which our products could provide them efficiencies and relief. They also did research into competitor's products. They read user guides, used demos, attended webinars. They spend time in conversations with other Product Owners to determine if cross product features could be useful in solving common stakeholder needs. If the Product Owner is to ensure that the work being done is going to satisfy stakeholder needs it seems that regular and continuous interactions with the stakeholders would be useful.